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Did you remove the Joomla 3 files before restoring the Joomla 4 backup?

Remember, this is already explained in the documentation, in a passage I wrote several years ago (it was first written back in 2010, when Joomla! 1.6 came out and slightly updated since):

Do not try to overwrite one Joomla! version family with a different one. Overwriting a major version with another (e.g. restoring a backup taken on Joomla! 3.7 on top of a site running Joomla! 2.5 or vice versa) or between different minor versions (e.g. restoring a backup taken on Joomla! 3.7 on top of a site running Joomla! 3.6 or vice versa) will NOT work. Joomla! moves files around between minor and major versions. Since the backup does not delete files not present in the backup archive this will end up with Joomla! being "confused" and malfunctioning. In these cases you should delete the existing files and folders (except, perhaps, user generated content) before restoring the backup. You can safely restore a sub-minor (path-level) version on top of another. For example, you can safely restore a Joomla! 3.7.5 site on top of a Joomla! 3.7.3 site or vice versa.

https://www.akeeba.com/documentation/akeeba-backup-joomla/restoring-backups.html#general-guidelines

So, here's what to do. Keep a copy of your backup archive files offline; delete the folders administrator, api, cache, cli, components, includes, language, layouts, libraries, media, modules, plugins, and templates; restore your backup; at the database restoration page select the Drop Same Prefix option for existing tables.

First, try deleting the .htaccess and .htpasswd files from your site's administrator folder — NOT the main folder! Directory password protection does not work the same on all servers. You might have a server which doesn't like the current .htpasswd file.

If this didn't help, something is wrong on the PHP side. Edit your configuration.php file. Change the following lines:

public $error_reporting = 'maximum';
public $debug = true;
public $gzip = false;

This turns on verbose error reporting which should give us more information about what is going on.

You don't get the popup because we removed the .htaccess and .htpasswd from the administrator folder which is what was giving you trouble.

Go to Password-protect Administrator in Admin Tools' main page to re-enable the password protection of your administrator folder.

It is possible that your server is not compatible with any of the password hashing methods we can use, or with the concept of password protecting directories. Both can be controlled at the web server configuration level (but cannot be overridden at the site level i.e. .htaccess file).

I can't see how. The code for the administrator password protection is identical in both Admin Tools 6 (Joomla 3) and Admin Tools 7 (Joomla 4). Joomla's version has nothing to do with it, it is implemented directly at the server level.

The only thing I can think of is that you had used different settings for the admin password protection on Joomla 3. Try changing the options for the admin password protection?

Therefore, it was definitely some setting in that feature that was changed compared to the defaults. Uninstalling and reinstalling reset the settings back to the default which is very conservative.